arXiv subjects

R. D. Gill

Publications and source records attributed to R. D. Gill.

17 recordsLinked to original sources

arXiv2023

Causal Inference for Complex Longitudinal Data: The Continuous Time g-Computation Formula

We extend Robins' theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the g-computation formula and a collection of powerful characterizations of the g-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past. We also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables: thus in a precise sense, these assumptions are "for free," or if you prefer, harmless.

math.ST↗

arXiv2022

Healthcare serial killer or coincidence? Statistical issues in investigation of suspected medical misconduct

Justice systems are sometimes called upon to evaluate cases in which health-care professionals are suspected of killing their patients illegally. These cases are difficult to evaluate because they involve at least two levels of uncertainty. Commonly in a murder case it is clear that a homicide has occurred, and investigators must resolve uncertainty about who is responsible. In the cases we examine here there is also uncertainty about whether homicide has occurred. Investigators need to consider whether the deaths that prompted the investigation could plausibly have occurred for reasons other than homicide, in addition to considering whether, if homicide was indeed the cause, the person under suspicion is responsible. In this report, prepared under the auspices of the Royal Statistical Society, we provide advice and guidance on the investigation and evaluation of such cases. Our work was prompted by concerns about the statistical challenges such cases pose for the legal system.

stat.AP↗

arXiv2022

Anna Karenina and The Two Envelopes Problem

The Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The Two Envelopes Problem (TEP) is a much-studied paradox in probability theory, mathematical economics, logic, and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasizes what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.

stat.OT↗

arXiv2022

Calculating LRs for presence of body fluids from mRNA assay data in mixtures

Messenger RNA (mRNA) profiling can identify body fluids present in a stain, yielding information on what activities could have taken place at a crime scene. To account for uncertainty in such identifications, recent work has focused on devising statistical models to allow for probabilistic statements on the presence of body fluids. A major hurdle for practical adoption is that evidentiary stains are likely to contain more than one body fluid and current models are ill-suited to analyse such mixtures. Here, we construct a likelihood ratio (LR) system that can handle mixtures, considering the hypotheses H1: the sample contains at least one of the body fluids of interest (and possibly other body fluids); H2: the sample contains none of the body fluids of interest (but possibly other body fluids). Thus, the LR-system outputs an LR-value for any combination of mRNA profile and set of body fluids of interest that are given as input. The calculation is based on an augmented dataset obtained by in silico mixing of real single body fluid mRNA profiles. These digital mixtures are used to construct a probabilistic classification method (a 'multi-label classifier'). The probabilities produced are subsequently used to calculate an LR, via calibration. We test a range of different classification methods from the field of machine learning, ways to preprocess the data and multi-label strategies for their performance on in silico mixed test data. Furthermore, we study their robustness to different assumptions on background levels of the body fluids. We find logistic regression works as well as more flexible classifiers, but shows higher robustness and better explainability. We test the system's performance on lab-generated mixture samples, and discuss practical usage in case work.

stat.AP↗

arXiv2010

A tight Tsirelson inequality for infinitely many outcomes

We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.

quant-ph↗

arXiv2008

Maximal violation of the CGLMP inequality for infinite dimensional states

We present a much simplified version of the CGLMP inequality for the 2 x 2 x d Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of d the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the 2 x 2 x inf scenario.

quant-ph↗

arXiv2006

Separable Measurement Estimation of Density Matrices and its Fidelity Gap with Collective Protocols

We show that there exists a gap between the performance of separable and collective measurements in qubit mixed-state estimation that persists in the large sample limit. We characterize such gap in terms of the corresponding bounds on the mean fidelity. We present an adaptive protocol that attains the separable-measurement bound. This (optimal separable) protocol uses von Neumann measurements and can be easily implemented with current technology.

quant-ph↗

arXiv2005

Optimal full estimation of qubit mixed states

We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number N of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary N, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large N. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.

quant-ph↗

arXiv2004

Estimating the causal effect of a time-varying treatment on time-to-event using structural nested failure time models

In this paper we review an approach to estimating the causal effect of a time-varying treatment on time to some event of interest. This approach is designed for the situation where the treatment may have been repeatedly adapted to patient characteristics, which themselves may also be time-dependent. In this situation the effect of the treatment cannot simply be estimated by conditioning on the patient characteristics, as these may themselves be indicators of the treatment effect. This so-called time-dependent confounding is typical in observational studies. We discuss a new class of failure time models, structural nested failure time models, which can be used to estimate the causal effect of a time-varying treatment, and present methods for estimating and testing the parameters of these models.

math.ST↗

arXiv2004

An invitation to quantum tomography (II)

The quantum state of a light beam can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We describe quantum tomography as an inverse statistical problem in which the state is the unknown parameter and the data is given by results of measurements performed on identical quantum systems. We present consistency results for Pattern Function Projection Estimators as well as for Sieve Maximum Likelihood Estimators for both the density matrix of the quantum state and its Wigner function. Finally we illustrate via simulated data the performance of the estimators. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation, studying other estimators, etc., and a main purpose of the paper is to bring these to the attention of the statistical community.

math.ST↗

arXiv2003

On Quantum Statistical Inference, II

Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, theoretical developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates a number of new concepts for an extension of classical statistical inference to the quantum context.

quant-ph↗

arXiv2003

On Quantum Statistical Inference, I

Recent developments in the mathematical foundations of quantum mechanics have brought the theory closer to that of classical probability and statistics. On the other hand, the unique character of quantum physics sets many of the questions addressed apart from those met classically in stochastics. Furthermore, concurrent advances in experimental techniques and in the theory of quantum computation have led to a strong interest in questions of quantum information, in particular in the sense of the amount of information about unknown parameters in given observational data or accessible through various possible types of measurements. This scenery is outlined (with an audience of statisticians and probabilists in mind).

quant-ph↗

arXiv2002

No time loophole in Bell's theorem; the Hess-Philipp model is non-local

Hess and Philipp have recently claimed that proofs of Bell's theorem have overlooked the possibility of time dependence in local hidden variables, hence the theorem has not been proven true. Moreover they present what is claimed to be a local realistic model of the EPR correlations. If this is true then Bell's theorem is not just unproven, but false. We refute both claims. Firstly we explain why time is not an issue in Bell's theorem, and secondly show that their hidden variables model violates Einstein separability. Hess and Philipp have overlooked the freedom of the experimenter to choose settings of a measurement apparatus at will: any setting could be in force during the same time period.

quant-ph↗

arXiv2002

Comment on "Exclusion of time in the theorem of Bell" by K. Hess and W. Philipp

A recent Letter by Hess and Philipp claims that Bell's theorem neglects the possibility of time-like dependence in local hidden variables, hence is not conclusive. Moreover the authors claim that they have constructed, in an earlier paper, a local realistic model of the EPR correlations. However, they themselves have neglected the experimenter's freedom to choose settings, while on the other hand, Bell's theorem can be formulated to cope with time-like dependence. This in itself proves that their toy model cannot satisfy local realism, but we also indicate where their proof of its local realistic nature fails.

quant-ph↗

arXiv2002

The Quantum Monty Hall Problem

We consider a quantum version of a well-known statistical decision problem, whose solution is, at first sight, counter-intuitive to many. In the quantum version a continuum of possible choices (rather than a finite set) has to be considered. It can be phrased as a two person game between a player P and a quiz master Q. Then P always has a strategy at least as good as in the classical case, while Q's best strategy results in a game having the same value as the classical game. We investigate the consequences of Q storing his information in classical or quantum ways. It turns out that Q's optimal strategy is to use a completely entangled quantum notepad, on which to encode his prior information.

quant-ph↗

arXiv2000

Fisher information in quantum statistics

Braunstein and Caves (1994) proposed to use Helstrom's {\em quantum information} number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing else than the maximal Fisher information in a measurement of the quantum system, maximized over all possible measurements. Combining this fact with classical statistical results, they argued that the quantum information determines the asymptotically optimal rate at which neighbouring states on some smooth curve can be distinguished, based on arbitrary measurements on $n$ identical copies of the given quantum system. We show that the measurement which maximizes the Fisher information typically depends on the true, unknown, state of the quantum system. We close the resulting loophole in the argument by showing that one can still achieve the same, optimal, rate of distinguishability, by a two stage adaptive measurement procedure. When we consider states lying not on a smooth curve, but on a manifold of higher dimension, the situation becomes much more complex. We show that the notion of ``distinguishability of close-by states'' depends strongly on the measurement resources one allows oneself, and on a further specification of the task at hand. The quantum information matrix no longer seems to play a central role.

quant-ph↗

arXiv1999

State estimation for large ensembles

We consider the problem of estimating the state of a large but finite number $N$ of identical quantum systems. In the limit of large $N$ the problem simplifies. In particular the only relevant measure of the quality of the estimation is the mean quadratic error matrix. Here we present a bound on the mean quadratic error which is a new quantum version of the Cramér-Rao inequality. This new bound expresses in a succinct way how in the quantum case one can trade information about one parameter for information about another parameter. The bound holds for arbitrary measurements on pure states, but only for separable measurements on mixed states--a striking example of non-locality without entanglement for mixed but not for pure states. Cramér-Rao bounds are generally derived under the assumption that the estimator is unbiased. We also prove that under additional regularity conditions our bound also holds for biased estimators. Finally we prove that when the unknown states belong to a 2 dimensional Hilbert space our quantum Cramér-Rao bound can always be attained and we provide an explicit measurement strategy that attains our bound. This therefore provides a complete solution to the problem of estimating as efficiently as possible the unknown state of a large ensemble of qubits in the same pure state. For qubits in the same mixed state, this also provides an optimal estimation strategy if one only considers separable measurements.

quant-ph↗